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Set-theoretic aspects of periodic FC-groups —

arXiv:math/9211201v1 [math.LO] 3 Nov 1992Set-theoretic aspects of periodic FC-groups —Extraspecial p-groups and Kurepa trees∗J¨org BrendleMathematisches Institutder Universit¨at T¨ubingenAuf der Morgenstelle 10W-7400 T¨ubingenGermanyAbstractWhen generalizing a characterization of centre-by-finite groups due to B. H. Neumann, M.J. Tomkinson asked the following question. Is there an FC-group G with |G/Z(G)| = κbut [G : NG(U)] < κ for all (abelian) subgroups U of G, where κ is an uncountable cardinal[16, Question 7A on p. 149].

We consider this question for κ = ω1 and κ = ω2. It turnsout that the answer is largely independent of ZFC (the usual axioms of set theory), andthat it differs greatly in the two cases.∗This work forms part of the author’s PhDThesis written under the supervision of Prof.Ulrich Felgner, T¨ubingen, 1991.1

IntroductionThe problem and its history. The purpose of this paper is to give some independenceproofs concerning the existence of FC-groups having some additional properties.

Recallthat a group G is FC iffevery element g ∈G has finitely many conjugates; i.e.iff[G : CG(g)] is finite for any g ∈G. We shall mostly be concerned with periodic FC-groups.In the fifties, B. H. Neumann gave the following characterization of centre-by-finitegroups; i.e.

groups with |G/Z(G)| < ω. (I) The following are equivalent for any group G.(i) G is centre-by-finite.

(ii) Each subgroup of G has only finitely many conjugates; i.e. [G : NG(U)] < ω forall U ≤G.If G is an FC-group both are equivalent to(iii) U/UG is finite for all U ≤G.Here UG denotes the largest normal subgroup of G contained in U, i.e.

UG := ∩g∈G g−1Ug;it is called the core of U in G. It was indicated by Eremin that in both (ii) and (iii) aboveit suffices to consider abelian subgroups (cf [16, 7.12(a) and 7.20]; also note that a groupsatisfying (iii) above is in general not FC [16, p. 142]).Following M. J. Tomkinson [15] (see also [4]), for an infinite cardinal κ, let Zκ denotethe class of groups G in which [G : CG(U)] < κ whenever U ≤G is generated by fewerthan κ elements (for κ > ω, this is equivalent to saying that [G : CG(U)] < κ for U ≤Gof size less than κ). Clearly Zω is just the class of FC-groups.

Generalizing Neumann’sresult Tomkinson proved in [15] (see also [16, Theorem 7.20]). (II) Let κ be an infinite cardinal.

The following are equivalent for any FC-group G in Zκ. (i) |G/Z(G)| < κ.

(ii) [G : NG(U)] < κ for all U ≤G. (iii) [G : NG(A)] < κ for all abelian A ≤G.

(iv) |U/UG| < κ for all U ≤G. (v) |A/AG| < κ for all abelian A ≤G.2

It was later shown by Faber and Tomkinson in [4] that the condition that G is FC can bedropped in (II).On the other hand one might ask whether the condition that G is Zκ is necessary toprove the equivalence of (i) through (v) in (II). Clearly the following implications alwayshold.

(i) =⇒(ii) =⇒(iii)(i) =⇒(iv) =⇒(v)But what about the others?One answer to this question was indicated by Tomkinson himself [16, p. 149]. Recallthat a p-group E is called extraspecial iffΦ(E) = E′ = Z(E) ∼= ZZ p. (Here Φ(E) is theFrattini subgroup of E; i.e.

the intersection of all maximal subgroups of E.) This impliesthat E/E′ is elementary abelian. Let E be an extraspecial p-group of size ω1 all of whoseabelian subgroups are countable (the existence of such groups was proved by S. Shelahand J. Stepr¯ans in [13] improving on earlier work of A. Ehrenfeucht and V. Faber [16,Theorem 3.12] who got the same result under the additional assumption of the continuumhypothesis (CH)).

Note that if U ≤E then U is either normal and so U = UE or U isabelian and so |U/UE| ≤|U| ≤ω. So for κ = ω1 and G = E, (iv) and (v) in (II) are true,whereas (i) is not.

Also, if A ≤E is maximal (abelian) with respect to A ∩E′ = 1, then⟨E′, A⟩= CE(A) = NE(A). Hence |A| = |NE(A)| = ω and [E : NE(A)] = ω1.

Thus (ii)and (iii) do not hold either.Why is this so? – To get (iv) and (v) but not (i) we used an extraspecial p-group suchthat all (maximal) abelian subgroups are small.

Dually, to get (ii) and (iii) but not (i) weshould use an extraspecial p-group such that all maximal abelian subgroups are large inthe sense that their indices are small. It will be one of our goals to discuss the existenceof such groups (see Theorems D and E below and § 5).

Tomkinson proved already in [14]that there are no such groups of size ω1 (this also follows from our more general TheoremC).The main results. For κ = ω1 our results are as follows.Theorem A.

Under CH there is an FC-group G with |G/Z(G)| = ω1 but [G :NG(A)] ≤ω for all abelian subgroups A ≤G.Theorem B. It is consistent (assuming the consistency of ZFC) that there is no3

FC-group G with |G/Z(G)| = ω1 but [G : NG(A)] ≤ω for all abelian subgroups A ≤G.Theorems A and B show that the question whether (i) and (iii) in (II) are equivalent forκ = ω1 is not decided by the axioms of set theory alone. The example used to proveTheorem A has a countable subgroup U with [G : NG(U)] = ω1.

So it does not answer thefollowingQuestion 1. Let κ = ω1.

Are (i) and (ii) in (II) equivalent for all FC-groups G?We conjecture that the answer is yes. Our reason for believing this is the following partialresult.Theorem C. Let κ = ω1.

Then (i) through (iii) in (II) are equivalent for all finite-by-abelian groups G.Here, a group G is called finite-by-abelian iffG′ is finite.Question 1 and Theorem C are very closely related to another problem of Tomkinson[16, Question 3F on p. 60]. Following [14] (see also [16, chapter 3]) let Z be the classof locally finite groups G satisfying: for all cardinals κ and all H ≤G of size < κ,[G : CG(H)] < κ.

So Z is the class of periodic groups in the intersection of the Zκ. AndY is the class of locally finite groups G satisfying: for all cardinals κ and all H ≤G ofsize < κ, [G : NG(H)] < κ.

Clearly Z ⊆Y. Tomkinson asked whether there are Y-groupswhich are not in Z.

He proved in [14, Theorem D(i)] that any extraspecial p-group in Yof size ω1 lies in Z. We generalize this by showingTheorem C’.

Y = Z for finite-by-abelian groups of size ω1.Theorem B’. Assuming the consistency of ZFC it is consistent that Y = Z forFC-groups of size ω1.The proofs of these results use the same ideas as the proofs of Theorem C and B, re-spectively, and we hope that our argument can be generalized to give a positive answertoQuestion 1’.

Is Y = Z for FC-groups of size ω1?Note that a positive answer to Question 1’ would give a positive answer to Question 1 too.For suppose there is a counterexample G. Then |G/Z(G)| = ω1 (and without loss we mayassume that |G| = ω1) but [G : NG(U)] ≤ω for all U ≤G. By Tomkinson’s result (II),G∈/Zω1, so G∈/Z, hence G∈/Y; i.e.

there is a countable U ≤G such that [G : NG(U)] = ω1,4

a contradiction. – The problem seems to be of group theoretical character, and mightinvolve a better understanding of countable periodic FC-groups.For κ = ω2 the picture changes considerably.

Recall that an uncountable cardinal κis (strongly) inaccessible iffit is regular (i.e., it is not the union of < κ sets of size < κ)and all cardinals λ < κ satisfy 2λ < κ (especially, κ is a limit cardinal). The existenceof inaccessible cardinals cannot be proved in ZFC; in fact, something much stronger istrue.

Let I denote the sentence there is an inaccessible cardinal. Then the consistency ofZFC can be proved in the system ZFC +I (see [9, chapter IV, Theorem 6.6 and p. 145]).Hence, by G¨odel’s Incompleteness Theorem, the consistency of ZFC +I cannot be provedfrom the consistency of ZFC alone.

– A weak Kurepa tree is a tree of height ω1 with ω2uncountable branches such that all levels have size ≤ω1. A Kurepa tree is a weak Kurepatree with countable levels (a more formal definition will be given in § 1).

The existence ofKurepa trees is consistent (assuming the consistency of ZFC), and the non-existence ofKurepa trees is equiconsistent with the existence of an inaccessible (see, again, our § 1 fordetails).Theorem D. Assume there is a Kurepa tree. Then there is an extraspecial p-groupof size ω2 such that [G : A] ≤ω1 for all maximal abelian subgroups A ≤G.On the other hand one can show (Theorem 5.4) that the existence of such a group impliesthe existence of a weak Kurepa tree.

In fact, we can prove the following much strongerresult.Theorem E. Assuming the consistency of ZFC + I it is consistent that for bothκ = ω1 and κ = ω2 and any FC-group G, (i) through (iii) in (II) are equivalent.We thus getCorollary. The following theories are equiconsistent.

(a) ZFC + I. (b) ZFC + for κ = ω1 and κ = ω2 and for any FC-group G: if |G/Z(G)| = κ then thereis an abelian subgroup A ≤G with [G : NG(A)] = κ.

(c) ZFC + any extraspecial p-group of size ω2 has an (abelian) subgroup with [G :NG(A)] = ω2.”(a) ⇒(b)” is Theorem E; ”(b) ⇒(c)” is trivial; and ”(c) ⇒(a)” follows from Theorem Dusing the equiconsistency concerning the non-existence of Kurepa trees mentioned above.It should be pointed out that we cannot prove the equivalence of (b) and (c); namely, it is5

consistent (assuming again the consistency of ZFC + I) that (c) holds but (b) does not(see Theorem 5.8). Still our Corollary shows again (cf [14] or [16, chapter 3, especially3.15]) the importance of the extraspecial p-groups in the class of periodic FC-groups.The organization of the paper.

Our results use mainly classical (modern) set theory.For algebraists who might not be familiar with this material, we give a short Introductionto this subject in § 1. We hope that this makes our work more intelligible.

The readerwho has seen forcing etc. before should skip the entire § 1.In the second section we show that a countable finite-by-abelian group is generatedby finitely many abelian subgroups (Theorem 2.2).

We also discuss what goes wrong whencountable is dropped from the assumption of the Theorem.In the third section we prove a result on automorphisms of countable periodic abeliangroups which turns out to be crucial for our arguments (Theorem 3.2); we will apply itin the proofs of Theorems B, C, and E. If we could generalize this result to countableperiodic FC-groups, we would get a positive answer to Questions 1 and 1’. Our originalproof involved a fragment of Ulm’s classification theorem.

Since then, M. J. Tomkinson hasfound a much shorter and more elegant proof which we reproduce with his permission...We close § 3 with the proof of Theorem C (and C’).Section 4 is devoted to the proofs of Theorems A and B (and B’); i.e. to the caseκ = ω1.It turns out that the knowledge of maximal abelian subgroups of countableperiodic FC-groups is essential.In section 5 we deal with the case κ = ω2 and the relationship between Kurepa treesand extraspecial p-groups; we prove Theorems D and E. We think that those results arethe most interesting and most beautiful of our work.We close with some generalizations in § 6.Finally note that we get most of the main results mentioned in the preceding subsectionas corollaries to more technical theorems and constructions, and we hope that the ideasinvolved in the latter might be useful when dealing with other problems as well.

They areTheorems 2.2, 3.2 and 4.4 (with its elaboration in 5.7) and the easy 5.4 (see also 5.7) –and the constructions in 4.2 (modified in 4.6 and 4.7) and 5.3 (modified in 5.9).Group-theoretic notation and basic facts on FC-groups. Our group-theoretic notation isstandard.

Good references are [12] for general group theory, [5] and [6] for abelian groups6

(which will be written additively), and [16] for FC-groups.For completeness’ sake we give our extension-theoretic notation. Let A, G be groups.If G ≤Aut(A), we let A×G denote the semidirect product of A and G. If τ : G2 →Ais a factor system (i.e.

∀g ∈G (τ(g, 1) = τ(1, g) = 1) and ∀f, g, h ∈G (τ(fg, h)τ(f, g) =τ(f, gh)τ(g, h))), we let E(τ) denote the corresponding extension (where the operation ofG on A is trivial). In the latter case, group multiplication is given by the formula∀(a, g), (b, h) ∈E(τ)(a, g) ∗(b, h) = (τ(g, h)ab, gh).More details can be found in [12, chapter 11].We note that in all of our results (in particular, in Theorems B, C, and E) it suffices toconsider periodic FC-groups.

The reason for this is as follows. By a result of ˇCernikov [16,Theorem 1.7], any FC-group can be embedded in a direct product of a periodic FC-groupand a torsion-free abelian group.

Now suppose G is an (arbitrary) counterexample to oneof our results; i.e. |G/Z(G)| = κ, but [G : NG(A)] < κ for all (abelian) A ≤G.

AssumeG ≤P ×T and π(G) = P, ρ(G) = T, where P is a periodic FC-group and T is torsion-freeabelian, and π and ρ are the projections. Clearly |P/Z(P)| = κ, and also [P : NP (A)] < κfor all (abelian) A ≤P.

This gives us a periodic counterexample.Acknowledgments. I should like to thank Ulrich Felgner and Frieder Haug for manystimulating discussions relating to the material of this work.

I am also grateful to M. J.Tomkinson for simplifying the proof of Theorem 3.2, and to both him and the referee formany valuable suggestions.§ 1. Set-theoretic preliminariesSet-theoretic Notation.

If X is a set, [X]κ denotes the set of subsets of X of size κ;[X]<κ is the set of subsets of X of size < κ; [X]≤κ etc. are defined similarly.

If X ∈[κ]nfor some n ∈ω, then X(i) (i < n) denotes the i-th element of X under the inheritedordering. Further set-theoretic notation can be found in [9] or [7].7

Delta-systems and almost disjoint sets.A family A of sets is called a delta-system(∆-system) if there is an R (called the root of A) such that∀A, B ∈A (A ̸= B =⇒A ∩B = R).The delta-system lemma [9, chapter II, Theorem 1.6] asserts that given a collection A ofsets of size < κ with |A| ≥θ where θ > κ is regular and satisfies ∀α < θ (|α<κ| < θ), thereis a B ⊆A of size θ which forms a ∆-system. We shall use it most often in case κ = ω.If κ is a cardinal, a family of sets A ⊆P(κ) is called almost disjoint (a.d.) iff∀A ∈A (|A| = κ) and ∀A, B ∈A (A ̸= B ⇒|A ∩B| < κ).Trees.

A tree is a partial order ⟨T, ≤⟩such that for each x ∈T, {y ∈T; y < x}is wellordered by <. Let T be a tree.

For x ∈T, the height of x in T (ht(x, T)) is theorder type of {y ∈T; y < x}. For each ordinal α, the α-th level of T is Levα(T) = {x ∈T; ht(x, T) = α}.

The height of T (ht(T)) is the least ordinal α such that Levα(T) = ∅.A branch of T is a maximal totally ordered subset of T.A weak Kurepa tree is a tree T of height ω1 with at least ω2 uncountable branchessuch that ∀α < ω1 (|Levα(T)| ≤ω1). Clearly, if CH holds, the complete binary treeof height ω1 is a weak Kurepa tree.

A Kurepa tree is a weak Kurepa tree T satisfying∀α < ω1 (|Levα(T)| ≤ω). A Kurepa family is an F ⊆P(ω1) such that |F| ≥ω2 and∀α < ω1 (|{A ∩α; A ∈F}| ≤ω).

It is easy to see [9, chapter II, Theorem 5.18] that thereis a Kurepa family iffthere is a Kurepa tree.Partial orders and forcing. Forcing was created by Cohen in the early sixties to solveCantor’s famous continuum problem; i.e.

to show that for any cardinal κ of cofinality > ωit is consistent that 2ω = κ – assuming the consistency of ZFC. Since then many otherindependence problems have been solved by the same method.

As forcing will occupy acentral position in our work, we briefly define its main notions. For a (very nicely written)introduction to this subject, we refer the reader to [9].Let ⟨IP, ≤⟩be a partial order (p.o.

for short; sometimes, IP will be referred to as forcingnotion). The elements of IP are called conditions.

If p, q ∈IP and p ≤q, then p is strongerthan q (or p is said to extend q). p and q are compatible iff∃r ∈IP (r ≤p ∧r ≤q); otherwisethey are incompatible (p⊥q).

A set D ⊆IP is called dense iff∀p ∈IP ∃q ≤p (q ∈D); D is8

open dense iffit is dense and ∀p ∈IP ∀q ∈D (p ≤q ⇒p ∈D). G ⊆IP is called a filter iff∀p, q ∈G ∃r ∈G (r ≤p ∧r ≤q) and ∀p ∈G ∀q ∈IP (p ≤q ⇒q ∈G).

– Now suppose Mis a countable transitive model for ZFC (called the ground model), and IP ∈M. A filterG ⊆IP is called IP-generic over M ifffor all dense D ∈M, G ∩D ̸= ∅.

The countability ofM implies that there exist always IP-generic G; also, if IP is non-trivial in the sense that∀p ∈IP ∃q, r ∈IP (q ≤p, r ≤p ∧q⊥r), then a IP-generic G cannot lie in M, and thegeneric extension M[G] (the smallest countable transitive model of ZFC containing Mand G) will be strictly larger than M. – The properties of M[G] can be described insideM using the forcing relation (∥−) as follows. For any object in M[G] there is a IP-namein M; we shall use symbols like ˘A, ˙A, ... to denote such names.

A sentence of the forcinglanguage is a ZFC-formula ψ with all free variables replaced by names. For such ψ andp ∈IP we write p ∥−IP ψ (p forces ψ) ifffor all G which are IP-generic over M, if p ∈G,then ψ is true in M[G].

The relation ∥−is definable in the ground model M. Furthermore,if G is IP-generic over M and ψ is true in M[G], then for some p ∈G, p ∥−IP ψ.An antichain in a p.o. IP is a pairwise incompatible set.

IP is said to satisfy the κ-cc(κ-chain condition, κ an uncountable cardinal) iffevery antichain A ⊆IP has size < κ.ccc (countable chain condition) is the same as ω1-cc. IP is κ-closed iffwhenever λ < κand {pξ; ξ < λ} is a decreasing sequence of elements in IP (i.e.

ξ < η ⇒pξ ≥pη), then∃q ∈IP ∀ξ < λ (q ≤pξ). A p.o.

IP preserves cardinals ≥κ (≤κ) iffwhenever G isIP-generic over M, and λ ≥κ (λ ≤κ, respectively) is a cardinal in the sense of M, it isalso a cardinal of M[G]. Cardinals which are not preserved are collapsed.

If IP has theκ-cc, then it preserves cardinals ≥κ, if it is κ-closed, it preserves cardinals ≤κ.A map e : IP →Q (where IP and Q are p.o.) is a dense embedding iff∀p, p′ ∈IP (p′ ≤p ⇒e(p′) ≤e(p)), ∀p, p′ ∈IP (p⊥p′ ⇒e(p)⊥e(p′)), and e(IP) is dense in Q .

Ife : IP →Q is dense, IP and Q are equivalent in the sense that they determine the samegeneric extensions. Any p.o.

can be embedded densely in a (unique) complete Booleanalgebra IB(IP) (the Boolean algebra associated with IP).Sometimes we want to repeat the generic extension process. This leads to the tech-nique of iterated forcing (see [1] or [8, chapter 2] for details).

We are mainly concernedwith two-step iterations which we shall denote by IP ∗˘Q .We set Fn(κ, λ, µ) := {p; p is a function, |p| < µ, dom(p) ⊆κ, ran(p) ⊆λ};Fn(κ, λ, µ) is ordered by p ≤q iffp ⊇q. Fn(κ, 2, λ) is called the ordering for adding9

κ Cohen subsets of λ; for λ = ω, the Cohen subsets are referred to as Cohen reals. Assumethat 2<λ = λ, λ regular; then Fn(κ, 2, λ) is λ-closed, has the λ+-cc, and so preserves cardi-nals.

Furthermore, if κλ = κ (in the ground model), then 2λ = κ in the generic extension.Cohen extensions can be split and thought of as a two-step iteration (cf [9, chapter VIII,Theorem 2.1] for the case λ = ω).For simplicity, we think of forcing as taking place over the whole universe V insteadof over a countable model M (though this is not correct from the formal point of view –see [9] for a discussion of this).Finally we come to internal forcing axioms. Those are combinatorial principles provedconsistent via iterated forcing; their statement captures much of this iteration.

The easiestis Martin’s Axiom MA. (MA) For all ccc p.o.

IP and any family D of < 2ω dense subsets of IP, there is a filter G inIP such that ∀D ∈D (G ∩D ̸= ∅).For the (rather involved) statement of the proper forcing axiom PFA we refer the readerto [2] or [8, chapter 3].Forcing and inner models.Sometimes the consistency of ZFC is not sufficient forproving the consistency of some combinatorial statement (C) via forcing, and one hasto start with a stronger theory (in general some large cardinal assumption) – e.g. theexistence of an inaccessible (ZFC + I).

In those cases we also want to show that the largecardinal assumption was really necessary; e.g. that Con(ZFC+C) implies Con(ZFC+I).The way this is usually done is by showing that if C holds in the universe V, then somecardinal is large (e.g.

inaccessible) in a sub-universe U (a transitive class model U ⊆Vsatisfying ZFC); such sub-universes are called inner models. The most important is theconstructible universe L, invented by G¨odel.To show the consistency of the non-existence of weak Kurepa trees, an inaccessible iscollapsed to ω2 (more correctly, the cardinals between ω1 and the inaccessible are collapsed)– see [11] or [1].

On the other hand, the non-existence of Kurepa trees in V implies thatω2 is an inaccessible cardinal in the sense of L (see [9, chapter VII, exercise (B9)]). Theconsistency of the existence of Kurepa trees can be proved by forcing or by showing thatthey exist in L.10

§ 2. The invariant g(G)2.1.

For any group G let g(G) – the generating number – denote the minimum numberof abelian subgroups of G needed to generate G. The following result should be thoughtof as generalizing the fact that any countable extraspecial p-group is a central sum ofextraspecial p-groups of order p3 [16, Corollary 3.10] – and so can be generated by twoabelian subgroups.2.2. Theorem.

For any countable finite-by-abelian group G, g(G) < ω.Proof. We make induction on |G′|.

The case |G′| = 1 is trivial. So suppose |G′| > 1.We set H := CG(G′).

As G is an FC-group, |G : H| < ω; so it suffices to show that His generated by finitely many abelian subgroups. H′ is a finite abelian group; i.e.

it is adirect sum of finite cyclic groups of prime power order: H′ = ⟨a0⟩⊕... ⊕⟨an⟩. There is aprime p and a natural number ℓsuch that o(an) = pℓ.

Let A := ⟨a0, ..., an−1, apn⟩. We shalldefine (recursively) two subgroups H0, H1 ≤H such that ⟨H0, H1⟩= H and H′k ≤A < H′for k ∈2.

Then the result follows by induction.Suppose H = {bm; m ∈ω}. Let m0 be minimal with the property that there is an msuch that [bm0, bm] ̸∈A.

Put b0, ..., bm0 into H0. Let c0 := bm0 and c1 := bm1 where m1is minimal such that [c0, bm1] ̸∈A, and put c1 into H1.

For m > m0, m ̸= m1 let d0m be aproduct of bm and powers of c0 and c1 such that [d0m, bk] ∈A for any k ∈m0+1∪{m1}. Wecontinue this construction recursively.

Suppose we are at step i; i.e. m2i, m2i+1, c2i, c2i+1and dim (m > m2i, m ̸= m2j+1 for j ≤i) have been defined.

Then let m2i+2 be minimalwith the property that there is an m such that [dim2i+2, dim] ̸∈A. Put dim2i+1, ..., dim2i+2into H0.Let c2i+2 := dim2i+2 and c2i+3 := dim2i+3 where m2i+3 is minimal such that[c2i+2, dim2i+3] ̸∈A, and put c2i+3 into H1.

For m > m2i+2, m ̸= m2j+1 for j ≤i + 1,let di+1mbe a product of dim and powers of c2i+2, c2i+3 such that [di+1m , dik] ∈A for anyk ∈((m2i, m2i+2] ∪{m2i+3}) −{m2j+1; j ≤i}.In the end H1 := ⟨c2j+1; j ∈ω⟩; and H0 is the group generated by the elements whichhave been put into H0. It is easy to see that H0 and H1 satisfy the requirements.(Remark.

The proof of this result is in two steps. The first shows that finite-by-abeliangroups are nilpotent of class 2-by-finite, and doesn’t require countability.

)11

This property of countable finite-by-abelian groups should be seen as correspondingto an old result of Baer’s, that a group G is centre-by-finite iffχ(G) < ω [16, Theorem7.4], where χ(G) denotes the minimum number of abelian subgroups needed to cover G.Nevertheless there are two drawbacks. First of all it is easy to construct a (countable)FC-group G with |G′| = ω but g(G) = 2.Secondly, our result doesn’t generalize tohigher cardinalities.

The important example of Shelah and Stepr¯ans [13] shows that thereare finite-by-abelian (even extraspecial) groups of size ω1 all of whose abelian subgroupsare countable. But even for nicer classes of groups there is nothing corresponding to theTheorem as is shown by the following2.3.

Example. Let E be the group generated by elements a, aα, α < ω1, satisfyingthe relations ap = apα = [a, aα] = 1 and [aα, aβ] = a for α < β. E is easily seen to be anextraspecial Z-group of exponent p. We will show that g(E) = ω1.For suppose that g(E) ≤ω.Then there are abelian subgroups An (n ∈ω) suchthat E is generated by the An.

Choose Γ ∈[ω1]ω1 and n ∈ω such that for all α ∈Γaα ∈⟨Ak; k < n⟩. For each such α and any k < n we can find bk,α ∈Ak such thataα = Qn−1k=0 bk,α (at least modulo a factor which is a power of a and which is irrelevantfor our calculation).

Now let Bk,α consist of the β so that aβ appears as a factor in bk,α.We may assume that the Bk,α form a delta-system with root Rk for any fixed k. LetCk,α := Bk,α −Rk. We can suppose that there is a jk such that |Ck,α| = jk, that for allα ∈Γ sup Rk < min Ck,α, that for α < β (both in Γ) sup Ck,α < min Ck,β, and that themultiplicities with which the aβ appear in the bk,α depend only on γ ∈Rk or i ∈jk (andnot on the specific α).

Thenbk,α =Yβ∈RkaℓββYi∈jkamiCk,α(i),where ℓβ, mi ∈p. An easy commutator calculation shows that the commutativity of Akimplies that Pi∈jk mi ≡O (mod p).

On the other hand,aα =n−1Yk=0bk,α =n−1Yk=0(Yβ∈RkaℓββYi∈jamiCk,α(i)).This equation cannot hold for any α with ({α} ∪∪k

Note. It is easy to see that E can be embedded in an extraspecial p-group F withg(F) = 2.

Namely, let F be the group generated by a, aα, bα (α < ω1) satisfying – inaddition to the above relations – bpα = [a, bα] = [bα, bβ] = 1 and[aα, bβ] =a−1if α < β,1otherwise.Then F = ⟨A0, A1⟩, where A0 = ⟨aαbα; α < ω1⟩and A1 = ⟨bα; α < ω1⟩. In fact, F is asemidirect extension of E.So the inequality g(G) ≤κ is not necessarily preserved when taking subgroups.

It ispreserved, however, when taking factor groups. This suggests that instead of dealing withg, one should consider the hereditary generating number hg(G) := sup{g(U); U ≤G}.

(A much easier example for this is the direct sum Eω of countably many extraspecialp-groups of size p3 (of exponent p for p > 2). g(Eω) = 2, but Eω contains the tree groupC of 4.2 which has g(C) = ω.)2.4.

Let QSDF be the QSD-closure of the class of finite groups; i.e. G ∈QSDF iffitis a factor group of a subgroup of a direct sum of finite groups.

QSDF is a subclass of Z[16, Lemma 3.7]. Tomkinson asked [16, Question 3F] whether Z ̸= QSDF.

This was shownto be true rather indirectly by Tomkinson and L. A. Kurdaˇcenko; namely Kurdaˇcenko [10,Theorem 4] proved that any extraspecial QSDF-group can be embedded in a direct sum ofgroups of order p3 with amalgamated centre, and Tomkinson gave a (rather complicated)example [16, Example 3.16] for an extraspecial Z-group which cannot be embedded in adirect sum of groups of order p3 with amalgamated centre.We shall show that the group E of 2.3 does not lie in QSDF, thus providing an easierexample. To this end, for any group G, let P(G) be the least cardinal κ such that anyset of pairwise non-commuting elements of G has size less than κ.

A canonical ∆-systemargument shows that G ∈QSDF implies P(G) ≤ω1 (this is a special instance of [3,Theorem 6]). On the other hand, the definition of E in 2.3 shows that P(E) = ω2.

HenceE ∈Z \ QSDF.13

§ 3. FC-automorphisms of countable periodic abelian groups3.1.

Let G be an FC-group. An automorphism φ of G is called FC-automorphism iff|{g (g−1)φ; g ∈G}| < ω; i.e.

iffthe semidirect extension of G by the group generated byφ is still an FC-group. For our discussion the following is important.3.2.

Theorem. Let A be a countable periodic abelian group.

Suppose Φ is a groupof FC-automorphisms of A with cf(|Φ|) > ω. Then there is a subgroup B ≤A such that|{Bφ; φ ∈Φ}| = |Φ|.Proof (Tomkinson).

First of all, for φ ∈Φ, let Aφ := ⟨a −aφ; a ∈A⟩. There are onlycountably many finite subgroups C ≤A.

If ΦC := {φ ∈Φ; Aφ ≤C}, then Φ = SC ΦC.Since cf(|Φ|) > ω, there is a C = C(A) ≤A such that |ΦC| = |Φ|. We make induction on|C|.Secondly we can restrict our attention to p-groups (for some fixed prime p).

Thegeneral result follows easily (as any periodic abelian group is the direct sum of its p-components which are characteristic subgroups).Now let m := exponent of C; i.e.m is the smallest integer such that pmC = 0.Then for all a ∈A of height ≥m and all φ ∈Φ, aφ = a. (To see this let a ∈Abe of height ≥m.Choose ˆa ∈A such that pmˆa = a.Let ˆb := ˆaφ −ˆa ∈C.Thenaφ = (pmˆa)φ = pm(ˆa + ˆb) = pmˆa = a.) Especially it suffices to consider reduced p-groups.As usual let A1 denote the subgroup of all elements of infinite height in A. Pr¨ufer’sTheorem [5, Theorem 17.3] says that A/A1 is a direct sum of cyclic p-groups.

By thepreceding paragraph, φ⌈A1 = id for any φ ∈Φ. Suppose A1 ∩C < C. Choose B < Acontaining A1 such that B/A1 ∩(C +A1)/A1 = 0 and [A : B] < ω (this is possible becauseA/A1 is a direct sum of finite groups).

Then either B satisfies the requirements of theTheorem, or B has as many automorphisms as A.In the latter case we are done byinduction because C(B) < C(A) = C.This shows that we may assume C ≤A1 (in particular, A1 ̸= 0). Now let D < C suchthat |C/D| = p. Each φ ∈Φ leaves D fixed and so induces an automorphism of A/D.

LetΦA/D be the group of induced automorphisms. If |ΦA/D| < |Φ|, then |ΦD| = |Φ|, and weare done by induction.14

So we may assume that |ΦA/D| = |Φ| and consider ¯A = A/D. There is an E ≤Asuch that E ∩C = D and A/E ∼= Cp∞(such an E can be constructed as follows: let ¯F bea complement of ¯C in {¯x ∈¯A; o(¯x) = p}; set ¯E := {¯x ∈¯A; if o(¯x) = pn then pn−1¯x ∈¯F};let E be the subgroup of A corresponding to ¯E).

For each ¯φ ∈ΦA/D, id⌈p ¯A = ¯φ⌈p ¯A. So¯A = ¯E + p ¯A implies that if ¯φ ̸= ¯ψ (¯φ, ¯ψ ∈ΦA/D) then ¯φ⌈¯E ̸= ¯ψ⌈¯E.

Hence ¯E ∩¯C = ¯0 givesus ¯E ¯φ ̸= ¯E ¯ψ. Therefore E has |ΦA/D| images under Φ.

This proves the Theorem.3.3. Proof of Theorems C and C’.

We have to show that for any finite-by-abeliangroup G of size ω1,(i) G ∈Z iffG ∈Y;(ii) |G/Z(G)| ≤ω iff[G : NG(A)] ≤ω for all abelian A ≤G.For suppose not. Then there is a finite-by-abelian group G which is not Z such thatin case (i): G ∈Y;in case (ii): [G : NG(A)] ≤ω for all abelian A ≤G.

(In case (ii), the fact that G is not in Z follows from Tomkinson’s result (II) mentionedin the Introduction.) Then G has a countable subgroup U with [G : CG(U)] = ω1.

LetV := U G := ⟨g−1Ug; g ∈G⟩.As G is FC, V ≤G is countable.By Theorem 2.2,g(V ) < ω, so there are n ∈ω and Ai ≤V abelian such that ⟨Ai; i < n⟩= V . ClearlyCG(V ) = ∩i

So either[G : NG(Ai)] = ω1 in which case we’re done, or [NG(Ai) : CG(Ai)] = ω1.In thatcase, we may assume G = NG(Ai), and G/CG(Ai) can be thought of as a group of FC-automorphisms of Ai, and we are in the situation of Theorem 3.2; i.e. we get a subgroupB ≤Ai such that [G : NG(B)] = ω1, a contradiction.3.4.

The argument in 3.3 shows that if one could prove the analogue of Theorem 3.2under the weaker assumption that A is FC instead of abelian, this would solve Questions1 and 1’ in the Introduction. So we should askQuestion 1”.

Suppose G is a countable periodic FC-group, and Φ is a group of FC-automorphisms of G with cf(|Φ|) > ω. Is there a subgroup U ≤G such that |{Uφ; φ ∈Φ}| = |Φ| ?15

§ 4. Maximal abelian subgroups of FC-groups4.1.

Maximal abelian subgroups (of FC-groups) are important for our discussion,especially those of countable periodic FC-groups in case κ = ω1. For Theorem A, we wantto construct a countable FC-group having an uncountable set of automorphisms such thaton each (maximal) abelian subgroup only countably many act differently (4.2 and 4.3).To prove Theorem B, we shall try to shoot a new abelian subgroup through an old setof automorphisms so that many of these automorphisms act differently on this group (4.4and 4.5).

These two procedures should be seen as being dual to each other (cf especially4.6). Therefore we pause for an instant to look at the lattice of abelian subgroups itself.Lemma.

If G is a Zκ-group with |G/Z(G)| ≥κ, then G has at least 2κ maximalabelian subgroups.Proof. We construct recursively a tree {Aσ; σ ∈2<κ} of subgroups of G with Z(G) ≤Aσ and |Aσ/Z(G)| < κ (Aσ/Z(G) is finitely generated in case κ = ω) as follows.

LetA⟨⟩:= Z(G). If α ∈κ is a limit ordinal and σ ∈2α, let Aσ := Sβ∈α Aσ⌈β.

So assumeα = β + 1 for some β ∈κ and σ ∈2β. Suppose CG(Aσ) is abelian.

Choose B ≤Gsuch that ⟨B, CG(Aσ)⟩= G and |B| < κ (or B is finitely generated if κ = ω). Then[G : CG(B)] < κ.

So [G : CG(B) ∩CG(Aσ)] < κ which contradicts |G/Z(G)| ≥κ. SoCG(Aσ) is non-abelian and there are g, h ∈CG(Aσ) such that [g, h] ̸= 1.Then setAσˆ⟨0⟩:= ⟨Aσ, g⟩and Aσˆ⟨1⟩:= ⟨Aσ, h⟩.In the end, for each f ∈2κ, extend Sσ⊂f Aσ to a maximal abelian subgroup Af.

Byconstruction, g ̸= f implies Ag ̸= Af.In fact, the proof of the Lemma shows that any abelian subgroup A with |AZ(G)/Z(G)| <κ is contained in at least 2κ distinct maximal abelian subgroups; and that it is containedin at least κ subgroups Bα, α < κ, with Z(G) ≤Bα and |Bα/Z(G)| < κ and which arepairwise incompatible in the sense that ⟨Bα, Bβ⟩is not abelian for α ̸= β – this fact willbe used in the proof of Theorem 4.4. below!As a consequence in case κ = ω we getCorollary. An FC-group has either finitely many or at least 2ω maximal abeliansubgroups.

It has finitely many iffit is centre-by-finite.16

4.2. As mentioned earlier we are concerned with the following problem.

Suppose G isan FC-group. Under which circumstances is there a set S of κ automorphisms of G suchthat for all abelian A ≤G, |{φ⌈A; φ ∈S}| < κ?

An easy necessary condition is g(G) ≥ω.We begin with the followingExample. For each n ∈ω we introduce a finite group Cn as follows.

Let An be anelementary abelian p-group of size pn, and Bn an elementary abelian p-group of size p(n2).We extend Bn by An with factor system τn as follows:τn(ai, aj) = 0if i ≥j,bh(i,j)otherwise,where h : [n]2 →n2is a bijection and the ai (bj, respectively) are generators of An (Bn).Let Cn be the extension (i.e. Cn = E(τn)).

Note that Cn is the free object on n generatorsin the variety of two-step nilpotent groups of exponent p (p > 2); and that it is a specialp-group with C′n = Φ(Cn) = Z(Cn) = Bn. Let C be the direct sum of the Cn.

If g isany function from ω to ∪An with g(n) ∈An, then g defines a maximal abelian subgroupMg := ⟨Bn, g(n); n ∈ω⟩. On the other hand each maximal abelian subgroup of C is ofthis form.

So the maximal abelian subgroups can be thought of as branches through atree. For later reference we shall therefore call C the tree group.Now assume CH.

Let {Mα; α < ω1} be an enumeration of the Mg. We introduce(recursively) a set of automorphisms {φα; α < ω1} of G := C ⊕D where D = ⟨d⟩is agroup of order p as follows: fix α; let {Nn; n ∈ω} be an enumeration of {Mβ; β < α};and let {ψn; n ∈ω} be an enumeration of {φβ; β < α}. We define φα and an auxiliaryfunction f : ω →ω recursively.

Suppose f⌈(n + 1) and φα⌈(Li

It is easy to see that {φα; α < ω1} is a set of (distinct) au-tomorphisms of G such that for all maximal abelian A ≤G, |{φα⌈A; α < ω1}| ≤ω.17

4.3. Proof of Theorem A.

Form the semidirect extension E of the group G definedin subsection 4.2 by the group H generated by the automorphisms {φα; α < ω1} (alsodefined in 4.2). Then E is easily seen to be an FC-group with the required properties (infact, for all abelian A ≤E, [E : CE(A)] ≤ω).4.4.

We now show that CH was necessary in the example above.Theorem.Let λ > κ+ be cardinals, κ regular.Denote by Fn(λ, 2, κ) the p.o.for adding λ Cohen subsets of κ.Suppose V |= 2<κ = κ.Then in V[G], where G isFn(λ, 2, κ)-generic over V, the following holds: for any Zκ-group H of size κ and anyset of automorphisms Φ of H of size > κ there is an abelian subgroup A ≤H such that|{φ⌈A; φ ∈Φ}| = |Φ|.Proof.Let H be any Zκ-group of size κ and with |H/Z(H)| = κ.Note that if|H/Z(H)| < κ, then Z(H) has the required property (as 2<κ = κ). We define the orderingIPH for shooting new abelian subgroups through H as follows: IPH := {A ≤H; A isabelian and A is generated by < κ elements }.

IPH is ordered by reverse inclusion; i.e.A ≤IPH B iffB ⊆A. IPH is κ-closed, and non-trivial by the discussion in 4.1.

Since2<κ = κ, IPH is trivially κ+ −cc. So forcing with IPH preserves cardinals and cofinalities.We first claim that if Φ is any set of automorphisms of H of size > κ in V, thenin V[G], where G is IPH-generic over V, there is an abelian subgroup A ≤H such that|{φ⌈A; φ ∈Φ}| = |Φ|.For A we take the generic object, i.e.

A = ∪{B ≤H; B ∈G}. Suppose the claim isfalse.

Let µ be regular with |Φ| ≥µ ≥κ+. Then there is a Ψ ⊆Φ in V[G] of size µ with∀φ, ψ ∈Ψ (φ⌈A = ψ⌈A).

So this statement is forced by a condition B ∈IPH; i.e. there isa IPH-name ˘Ψ such thatB ∥−˘Ψ ⊆Φ ∧|˘Ψ| = µ ∧∀φ, ψ ∈˘Ψ (φ⌈˘A = ψ⌈˘A).As |IPH| = 2<κ = κ, there is (in V) a X ∈[Φ]µ and a C ≤IPH B such thatC ∥−X ⊆˘Ψ.Now, C is an abelian subgroup of the Zκ-group H of size less than κ.

So [H : CH(C)] < κ.As |X| > κ and 2<κ = κ, |{χ⌈CH(C); χ ∈X}| = µ so that we can find ψ, χ ∈X andc ∈CH(C) \ C such that ψ(c) ̸= χ(c). But then the condition ⟨C, c⟩forces contradictorystatements.

This proves the claim.18

Next we remark that for any Zκ-group H of size κ, IPH is equivalent (from theforcing theoretic point of view) to the Cohen forcing Fn(κ, 2, κ) for adding a single newsubset of κ.For κ = ω this follows from the fact that any two non-trivial countablenotions of forcing are equivalent [9, chapter VII, exercise (C4), p. 242]. The proof forthis generalizes as follows.

Let {Aα; α < κ} enumerate IPH. We construct recursivelya dense embedding e ¿from {p ∈Fn(κ, κ, κ); dom(p) ∈κ} into IPH.

Let α < κ andsuppose e⌈{p ∈Fn(κ, κ, κ); dom(p) ∈α} has been defined. If α is limit let, for any pwith dom(p) = α, e(p) = Sβ<α e(p⌈β).

So suppose α = β + 1 for some β ∈κ. There isby induction (at least) one p0 ∈Fn(κ, κ, κ) with dom(p0) = β such that Aβ is compatiblewith e(p0).

For each p ∈Fn(κ, κ, κ) with dom(p) = β choose a maximal antichain Mp ofsize κ of conditions below e(p) in IPH (the existence of such an antichain is guaranteedby the discussion in 4.1), such that ⟨Aβ, e(p0)⟩is a subgroup of some group in Mp0. Lete⌈{q ∈Fn(κ, κ, κ); dom(q) = α and q⌈β = p} be a bijection onto Mp.

It is easy to checkthat e works.The same argument shows that {p ∈Fn(κ, κ, κ); dom(p) ∈κ} can bedensely embedded into Fn(κ, 2, κ). This gives equivalence (cf § 1).Finally we prove the Theorem.

Let H and Φ be as in the statement of the Theorem.First suppose |Φ| < λ. Then Φ is contained in an initial segment of the extension, andany subset which is Cohen over this initial segment produces the required A by the abovearguments.

So suppose |Φ| ≥λ. In that case we think of the whole extension as a two-stepextension which first adds λ and then µ Cohen subsets of κ, where λ ≥µ ≥κ+ is regularwith cf(|Φ|) ̸= µ and |Φ| > µ.

Then there is a subset Ψ ∈[Φ]|Φ| which is contained in aninitial segment of the second extension, and our argument applies again.Remark. Note that in the Theorem, the assumption |H| = κ may be replaced by|H/Z(H)| = κ.

The p.o. IPH in the proof contains in that case the abelian subgroups Awith Z(H) ≤A and |A/Z(H)| < κ.4.5.

Proof of Theorems B and B’. Let V |= ZFC.

We show that in the model obtainedby adding ω2 Cohen reals to V,(i) any Y-group of size ω1 is a Z-group;(ii) there is no FC-group G with |G/Z(G)| = ω1 but [G : NG(A)] ≤ω for all abeliansubgroups A ≤G.For suppose not. Then there is an FC-group G of size ω1 which is not in Z such that19

in case (i): G ∈Y;in case (ii): [G : NG(A)] ≤ω for all abelian A ≤G.We argue as in the proof of Theorems C and C’ (subsection 3.3) using 4.4 instead of 2.2:let U ≤G be countable with [G : CG(U)] = ω1; let V := U G. Apply 4.4 (with κ = ω,λ = ω2, H = V and Φ = G/CG(V )) to get an abelian A ≤V with [G : CG(A)] = ω1. Nowfinish as in 3.3 with Theorem 3.2.4.6.

The proof of Theorem B shows that its statement follows from MA + 2ω > ω1.Still this is not the right way to look at the problem from the combinatorial point of view.Namely, when iterating Cohen forcing one merely goes through one particular ccc p. o.,whereas MA asserts that generic objects exist for all ccc p. o. – not only for those whichshoot new abelian subgroups through an FC-group G but also for those which shoot anew automorphism through G (see below).

The consequences of this will become clear in§ 5 (see the difference between Theorems E and 5.8).Also MA is a weakening of CH, and many statements which are provable in ZFC +CH are still provable in ZFC + MA if we replace ω by < 2ω. We shall see now that thisis the case for our problem as well.Proposition.

Assume MA. Then there is an FC-group G with |G/Z(G)| = 2ω but[G : NG(A)] < 2ω for all abelian subgroups A ≤G.Proof.

Let C be again the tree group of 4.2. We define the partial order Q C forshooting new automorphisms through C ⊕D (where D = ⟨d⟩is again a group of orderp).

Q C := {(φ, A); φ is a finite partial automorphism of C ⊕D with (i) ∃n ∈ω withdom(φ) = Li∈n Ci⊕D, (ii) ∀c ∈dom(φ) ∃k ∈p (cφ = c+kd) and (iii) φ⌈(Li∈n Bi⊕D) =id; and A is a finite collection of maximal abelian subgroups of C }; (φ, A) ≤Q C (ψ, B) iffφ ⊇ψ and A ⊇B and ∀c ∈(dom(φ)−dom(ψ))∩(∪B) (cφ = c). Q C is ccc and genericallyshoots a new automorphism through C ⊕D which equals the identity on all old abeliansubgroups from some point on.To prove the Proposition let {Aα; α < 2ω} enumerate the maximal abelian subgroupsof C. We construct recursively a set of automorphisms {φα; α < 2ω}.

Let Dα := {Dβ; β <α} where Dβ := {(φ, A) ∈Q C; Aβ ∈A and cφ ̸= cφβ for some c ∈dom(φ)}. Each Dβ isdense in Q C; hence, by MA, there is a Dα-generic filter Gα.

Let φα := ∪{φ; ∃A (φ, A) ∈Gα}.Then for all maximal abelian A ≤C ⊕D, |{φα⌈A; α < 2ω}| < 2ω.Now let20

G := (C ⊕D)×⟨φα; α ∈2ω⟩.Certainly one should ask whether MA is necessary at all in the above result; orwhether it can be proved in ZFC alone.It turns out that the answer (to the secondquestion) is no, at least if we assume the existence of an inaccessible cardinal – see § 5(Theorem E).4.7. It is quite usual that combinatorial statements are not decided by ¬CH.

Againthis is true in our situation.Proposition.It is consistent that 2ω > ω1 and there is an FC-group G with|G/Z(G)| = ω1, but [G : NG(A)] ≤ω for all abelian subgroups A ≤G.Sketch of the proof. The proof uses the tree group of 4.2 as main ingredient.

Startwith V |= 2ω > ω1 and make a finite support iteration of length ω1 of the partial orderQ C described in 4.6.21

§ 5. Extraspecial p-groups and Kurepa trees5.1.

The goal of this section is a detailed investigation of extraspecial p-groups, es-pecially of those of size ω2.The philosophy behind this is that many bad things thatcan happen to (periodic) FC-groups already happen in case of extraspecial p-groups; oreven that bad periodic FC-groups involve bad extraspecial groups – the most surprisingexample for this is Tomkinson’s result that a periodic FC-group G which does not lie in Ycontains U≤V ≤G such that V/U is extraspecial and not in Y (see [14] or [16, Theorem3.15]). Our main contribution in this direction is the equiconsistency result mentioned inthe Introduction (Corollary to Theorems D and E).

Another example is the equivalence inProposition 5.5. – On the other hand, because of their simple algebraic structure (e.g., thefact that subgroups are either normal or abelian), extraspecial examples are in general theeasiest to construct, and such constructions depend only on the underlying combinatorialstructure – the classical example for this is the existence of a Shelah-Stepr¯ans group [13].We let Yκ be the class of groups in which [G : NG(U)] < κ whenever U ≤G isgenerated by fewer than κ elements.

So the class Y is just the class of locally finite groupsin the intersection of the Yκ; and also Yω = Zω = the class of FC-groups. It follows fromTheorems B’ and C’ that Yω1 and Zω1 are consistently equal for periodic FC-groups, andthat they are equal for periodic finite-by-abelian groups.By Tomkinson’s result mentioned in the Introduction (II), if λ < κ are cardinals andG is a group with |G/Z(G)| = κ and [G : NG(U)] ≤λ for all subgroups U ≤G, then Gis Yµ for any µ ≥λ+ but not Zκ.

Furthermore, for extraspecial p-groups G, the followingare equivalent (where κ is any cardinal). (i) [G : NG(A)] < κ for all (abelian) subgroups A ≤G.

(ii) For all maximal abelian subgroups A of G, [G : A] < κ.In particular, an extraspecial p-group of size ω2 whose maximal abelian subgroups satisfy[G : A] ≤ω1 is Yω2 but not Zω2.This should motivate us to study the three classes Yω1 = Zω1, Yω2, and Zω2 forextraspecial p-groups more thoroughly. Clearly, there are groups lying in none or in allof these classes, or in Zω2 \ Zω1.

The existence of groups which are in Yω2 \ Zω2 or inZω1 \ Zω2 will be discussed in the subsequent subsections (up to 5.5). Our results can be22

summarized in the following chart.Zω1=Yω1¬Zω1=¬Yω1Zω2easyeasy5.3.and 5.4. (follows from the¬Zω2 but Yω2?(cf.5.5.

)existence of Kurepa trees, and impliesthe existence of weak Kurepa trees)¬Yω25.5. (equivalent to theeasyexistence of Kurepa trees)5.2.

The following is useful for the proof of Theorem D.Lemma (Folklore). Assume there is a Kurepa family.

Then there is an a. d. Kurepafamily of the same size.Proof. Let {Aα; α < κ} be a Kurepa family (where κ ≥ω2).

Let f be a bijectionbetween {Aα ∩β; α < κ, β < ω1} and ω1. Then {{f(Aα ∩β); β < ω1}; α < κ} is easilyseen to be an a. d. Kurepa family.5.3.

Proof of Theorem D. Let E be a Shelah-Stepr¯ans-group [13] of size ω1, and letA = {Aα; α < ω2} be an a. d. Kurepa family. We extend E semidirectly by an elementaryabelian group B of automorphisms using A as follows: for all α < ω2 define φα byaβφα = aβif β = 0 or β ̸∈Aα,aβa0otherwise,where a0 generates Z(E) and {aβ; 1 ≤β < ω1} generates E. Set B := ⟨φα; α < ω2⟩.This completes the construction.

G := E×B is easily seen to be extraspecial.Now suppose A ≤G is abelian. Let π(A) denote the subgroup of E generated bythe projection of A on the first coordinate (we think of the semidirect product as a set oftuples).

We claim that π(A) is countable. For suppose not.

Then clearly a0 ∈π(A). LetC be a maximal abelian subgroup of π(A).

C is countable, and Cπ(A)(C) = C. Choose a23

subset {(bα, ψα); α < ω1} of A such that C ≤⟨bn; n ∈ω⟩and bα ̸= bβ for α ̸= β. Nowlet Bα consist of the β so that aβ appears as a factor in bα. We may assume that theBα (α ≥ω) form a delta-system with root R. Let Cα := Bα \ R. We can suppose thatthere is a j such that |Cα| = j for α ≥ω, that for α < β we have sup Cα < min Cβ, andthat the multiplicities with which the aβ appear in the bα depend only on γ ∈R or i ∈j.As A is a. d., we may assume that for each of the (countably many) automorphisms φδappearing as a factor in some ψn (n ∈ω) and each i ∈j either ∀α ≥ω (aCα(i)φδ = aCα(i))or ∀α ≥ω (aCα(i)φδ = aCα(i)a0) (without loss the corresponding Aδ’s are disjoint aboveCω(0)).

In particular we have that for fixed n ∈ω, cn := b−1α (bα)ψn = b−1β(bβ)ψn for anyα, β ≥ω. As A is a Kurepa family, we may assume that ψα⌈C = ψβ⌈C for any α, β ≥ω.But then[(bn, ψn), (bα, ψα)] = ((b−1n ψ−1n )ψ−1α(b−1α )ψ−1α , ψ−1n ψ−1α ) (bnψα bα, ψnψα)= (b−1n(b−1α )ψn (bn)ψα bα, 1) = (c−1n dn[bn, bα], 1),where dn = bnψα b−1n .

As C is maximal abelian in π(A), there is certainly an n ∈ω suchthat [bn, bα] ̸= [bn, bβ] for some α, β ≥ω. But then the above calculation shows that Acannot be abelian.Now the fact that π(A) is countable and that A is a Kurepa family implies [G :CG(π(A))] ≤ω1 (in fact, equality holds unless π(A) is finite, because E is a Shelah-Stepr¯ans-group).

If ρ(A) is the projection of A on the second coordinate, [G : CG(ρ(A))] ≤ω1 holds trivially; and A ≤⟨π(A), ρ(A)⟩implies [G : NG(A)] ≤[G : CG(A)] ≤ω1.5.4. Theorem.

If there is an extraspecial p-group of size ω2 in Yω2 but not in Zω2,then there is a weak Kurepa tree.Proof. Let G be such a group.

Choose U ≤G of size ω1 such that [G : CG(U)] = ω2.Let {uα; α < ω1} generate U. Let {fβ; β < ω2} be a subset of G\U such that fαCG(U) ̸=fβCG(U).

Define gβ : ω1 →p for β < ω2 by gβ(α) = k iff[uα, fβ] = ka, where a generatesG′. We claim that the gβ form the branches of a weak Kurepa tree.For suppose not.

Then there is an α ∈ω1 such that |{gβ⌈α; β < ω2}| = ω2. Thisimmediately implies that [G : CG(V )] = ω2 for a countable subgroup V ≤U.

V is a directsum of an extraspecial and an abelian group; especially g(V ) ≤2 (this follows from thefact that countable extraspecial p-groups are central sums of groups of order p3). So there24

is an abelian A ≤V such that [G : CG(A)] = ω2. Cutting away G′ if necessary we mayassume that [G : NG(A)] = ω2, contradicting the fact that G ∈Yω2.Note that in the hypothesis of the Theorem, extraspecial p-group can be replaced by(periodic) finite-by-abelian group.

To see that this more general result is true, just applyTheorems 2.2 and 3.2 at the end of the proof. And if Question 1” had a positive answer,we could prove this for (periodic) FC-groups.There is a gap between Theorem D and Theorem 5.4.We feel that it should bepossible to make a construction like the one in 5.3 using a weak Kurepa tree only.5.5.

Using the same techniques as in 5.3 and 5.4 we getProposition. The following are equivalent.

(i) There is a Kurepa tree. (ii) There is an extraspecial p-group which is Zω1 but not Zω2.

(iii) There is an FC-group which is Zω1 but not Zω2.Proof. To see one direction ( (i) ⇒(ii) ) let E be any extraspecial Z-group of sizeω1, and let, as in 5.3, A = {Aα; α < ω2} be an a.d. Kurepa family.

For all α < ω2 defineφα byaβφα = aβif β = 0 or β ̸∈Aα,aβa0otherwise,where a0 generates Z(E) and {aβ; 1 ≤β < ω1} generates E. Set G := E×⟨φα; α < ω2⟩.Clearly G has the required properties.Conversely, to see (iii) ⇒(i), make the same construction as in 5.4.In general, the group constructed in the first part of the proof will not lie in Yω2either. Hence the only question left open is whether there are extraspecial Zω1-groups inYω2 \ Zω2.

We conjecture that they exist in the constructible universe L. Such a group ofsize ω2 would lie in Y \ Z as well and so give an answer to Question 3F in [16].On the other hand, unlike the other classes considered so far, the consistency of ZFCalone implies the consistency of the non-existence of extraspecial Zω1-groups in Yω2 \ Zω2.To see this, let V |= ZFC + GCH. Add ω3 Cohen subsets of ω1.

We claim that in theresulting model V[G], there are no such groups. For suppose G is such a group.

Find U ≤Gof size ω1 with [G : CG(U)] = ω2, without loss U≤G. Apply 4.4 with λ = ω3, κ = ω1,25

H = U, and Φ = G/CG(U) (this can be done as U ∈Zω1). Find A ≤U abelian suchthat [G : CG(A)] = ω2.

Cutting G′ away, if necessary, we can assume CG(A) = NG(A), acontradiction.5.6. We now want to turn to the proof of Theorem E. Certainly, in a model where itsstatement is true, neither of the bad situations discussed in 4.2 (and 4.3, 4.6, 4.7) and in5.3 can occur.

So we’d better look for a model where there are no Kurepa trees and whereCH is false. The discussion in 5.4 and 4.4 suggests that there shouldn’t be weak Kurepatrees either and that there should be reals Cohen over L. One possible attack would beto destroy all weak Kurepa trees by collapsing an inaccessible to ω2 (as in [11, §§3,4] or[1, §8]) and then to add ω2 Cohen reals (by the last ω2 we mean, of course, the ω2 ofthe intermediate model).

Unfortunately, we don’t know whether it is true in general thatthere are no weak Kurepa trees in the final extension; but this is true if the intermediatemodel is Mitchell’s [11]. In fact, it turns out that in this case the second extension isunnecessary, and what we want to show consistent already holds in Mitchell’s model.

Thereason for this is essentially that this model is gotten by first adding κ Cohen reals (whereκ is inaccessible) and then collapsing κ to ω2 using a forcing which does not add reals –and hence does not destroy the nice situation created by the Cohen reals – while killingall weak Kurepa trees.First we will review Mitchell’s model and some elementary facts about it. Let V |=”ZFC+GCH+ there is an inaccessible”.

Let κ be inaccessible in V. IP = Fn(κ, 2, ω) is the order-ing for adding κ Cohen reals. IB = IB(IP) is the Boolean algebra associated with IP.

SetIPα := {p ∈IP; supp(p) ⊆α} for α < κ. IBα is the Boolean algebra associated with IPα.f ∈V is in the set A of acceptable functions iff(1) dom(f) ⊆κ; ran(f) ⊆IB;(2) |dom(f)| ≤ω;(3) f(γ) ∈IBγ+ω for γ ∈dom(f).If F is IP-generic over V, f ∈A, then we define ¯f : dom(f) →2 in V[F] by ¯f(γ) = 1 ifff(γ) ∈F. Define Q in V[F] by letting the underlying set of Q be A and f ≤Q g iff¯f ⊇¯g.

So we get a 2-step iteration IP ∗Q with (p, f) ≤(q, g) iffp ≤q and p ∥−IPf ≤Q g.We shall denote the final extension by V[F][G].Facts (Mitchell [11]). (1) Suppose p ∥−IP” ˘D is open dense in Q below f”, wheref ∈A.

Let g ∈A such that g ⊇f. Then there is h ∈A such that h ⊇g and p ∥−IPh ∈˘D.26

(2) Q does not add new functions with countable domain over V[F]; i.e. if t : ω →V[F] where t ∈V[F][G], then t ∈V[F].

(3) Let {gα; α < κ} ⊆A. Then there are X ∈[κ]κ and g ∈A such that ∀α, β ∈X (dom(gα) ∩dom(gβ) = dom(g)) and ∀α ∈X (gα⌈dom(g) = g).

(4) IP∗Q preserves ω1 (this follows from the ccc-ness of IP and fact (2)) and cardinals≥κ (it follows from (3) that IP ∗Q is κ-cc), but collapses all cardinals in between to ω1;i.e. κV = ωV[F][G]2.

(5) In V[F][G], 2ω = 2ω1 = ω2. (6) Let ν < κ be such that ν′ + ω ≤ν for each ν′ < ν.

Then the generic extensionvia IP ∗Q can be split in a 2-step extension, the first of which adds |ν| Cohen reals and aQ ⌈ν-generic function from ν to 2, whereas the second adds the remaining κ Cohen realsand the remaining part of the Q -generic function ¿from κ to 2. (7) In V[F][G], there are no weak Kurepa trees.Proofs.

(1) to (5) are (more or less) 3.1 to 3.5 in [11]; concerning (3) we note that itis proved via a straightforward ∆-system-argument. (6) is made more explicit on pp.

29and 30 in [11] and proved in 3.6. For (7), see 4.7 in [11].5.7.

Proof of Theorem E. We show that V[F][G] |=”for both ω1 and ω2 and any FC-group G, (i) through (iii) in (II) are equivalent”, where V[F][G] is Mitchell’s model as inthe preceding section.Counterexamples G with |G/Z(G)| = ω1 are easily excluded. Without loss such Gwould have size ω1.

By the κ-cc of IP ∗Q (which follows from the ccc of IP and fact (3)in 5.6) it would lie in an intermediate extension (see fact (6)). Any real Cohen over thisintermediate extension shows that the assumption was false (by the argument of Theorem4.4).Suppose G is a counterexample with |G/Z(G)| = ω2; without loss |G| = ω2; [G :NG(A)] ≤ω1 for all abelian A ≤G; and there is a U ≤G of size ≤ω1 such that[G : CG(U)] = ω2 (because G cannot be a Zω2-group by Tomkinson’s result (II) in theIntroduction).

Without loss |U| = ω1. Let V := U G = ⟨x−1Ux; x ∈G⟩.

As G is anFC-group, |V | = ω1; and V ≤G. Clearly |G/CG(V )| = ω2.

For any ¯g ∈G/CG(V ) definea function f¯g : V →V by f¯g(v) := g−1vgv−1 where g ∈¯g is arbitrary. Think of {f¯g; ¯g ∈G/CG(V )} as the set of branches through a tree T. As V[F][G] does not contain weakKurepa trees (fact (7) in 5.6), there is a countable S ⊆V such that {f¯g⌈S; ¯G ∈G/CG(V )}27

has size ω2. Let W := ⟨SG⟩.

W is a countable normal subgroup of G; and |G/CG(W)| = ω2by construction.We now want to prove that there is an abelian A ≤W such that [G : CG(A)] = ω2(main claim). The way we do this is an elaboration of the proof of Theorem 4.4.

For thisargument it is crucial that we use Mitchell’s model and not just any model without weakKurepa trees.We think of G/CG(W) as a group of automorphisms Φ of W; more explicitly, Φ : ω2 →Aut(W). In V[F], let ˘Φ be a Q -name for Φ.

Let Dα (α < κ) be the set of conditionsdeciding ˘Φ(α). Dα is open dense by fact (2).

Let ˙Dα be a IP-name for Dα (α < κ). Then∥−IP” ˙Dα is open dense”.By fact (1) there are gα ∈A such that∥−IP gα ∈˙Dα.So in V[F], there are φα such that gα ∥−Q ˘Φ(α) = φα.

Let ˙φα (α < κ) be IP-names forthe φα. Then ∥−IP gα ∥−Q ˙Φ(α) = ˙φα, where ˙Φ is a IP-name for ˘Φ.

Using fact (3) we getX ∈[κ]κ and g ∈A such that ∀α, β ∈X, dom(gα) ∩dom(gβ) = dom(g) and ∀α ∈X,gα⌈dom(g) = g. Now we split IP into two parts (i.e. IP = IP1 × IP2) such that(1) IP1 adds κ Cohen reals and IP2 adds one Cohen real;(2) there is Y ∈[X]κ such that ∀α ∈Y (φα ∈V[F1]) where F1 is IP1-generic over V.So in V[F1],∥−IP2 gα ∥−Q ˙Φ(α) = φα,where α ∈Y .

From now on we work in V[F1]. As in the proof of Theorem 4.4 we think ofIP2 as adding a new abelian subgroup of W. Let ˙A be a IP2-name for this generic object.The rest of the proof of the main claim is by contradiction.

Suppose that∥−IP∗Q ∀B ≤W abelian (|{ ˙Φ(α)⌈B; α < κ}| < κ).Especially, in V[F1],∥−IP2∗Q |{ ˙Φ(α)⌈˙A; α ∈Y }| < κ.Hence,∥−IP2∗Q ∃α ∀β ≥α (β ∈Y ⇒∃γ < α (γ ∈Y ∧˙Φ(β)⌈˙A = ˙Φ(γ)⌈˙A)).28

So there are α < κ and C ∈IP2, h ⊇g, h ∈A such that(C, h) ∥−IP2∗Q ∀β ≥α (β ∈Y ⇒∃γ < α (γ ∈Y ∧˙Φ(β)⌈˙A = ˙Φ(γ)⌈˙A)).Choose Z ∈[Y \ α]κ such that for β ∈Z, gβ and h are compatible (in V). For β ∈Z let(in V[F]) Dβ be the set of conditions forcing ˘Φ(β)⌈A = ˘Φ(γ)⌈A for some γ < α. Dβ isopen dense below h. Let ˙Dβ be a IP2-name for Dβ.

By fact (1), there are hβ such thathβ ⊇gβ and hβ ⊇h andC ∥−IP2 hβ ∈˙Dβ.I.e. (C, hβ) ∥−IP2∗Q ˙Φ(β)⌈˙A = φβ⌈˙A = ˙Φ(γ)⌈˙Afor some γ = γ(β) < α.

Choose by fact (3) Z′ ∈[Z]κ and ˜h ⊇h (˜h ∈A of course) and γsuch that(1) ∀β1 ̸= β2 ∈Z′ (dom(hβ1) ∩dom(hβ2) = dom(˜h)); ∀β ∈Z′ (˜h = hβ⌈dom(˜h));(2) γ(β) = γ for β ∈Z′.C ≤W is a finite abelian subgroup. So [W : CW (C)] < ω.

As |Z′| = κ, |{φβ⌈CW (C); β ∈Z′}| = κ so that we can find c ∈CW (C) \ C and β1, β2 ∈Z′ such that (c)φβ1 ̸= (c)φβ2.Then(⟨C, c⟩, hβ1 ∪hβ2) ∥−IP2∗Q ” ˙Φ(β1)⌈˙A = φβ1⌈˙A = ˙Φ(γ)⌈˙A = φβ2⌈˙A = ˙Φ(β2)⌈˙Aand φβ1⌈˙A ̸= φβ2⌈˙A, ”which is a contradiction.This ends the proof of the main claim and shows that there is indeed an abelianA ≤W such that [G : CG(A)] = ω2. Then either [G : NG(A)] = ω2 or we apply Theorem3.2 – as we did before in the proofs of Theorems B and C – to get B ≤A such that[G : NG(B)] = ω2.

This is the final contradiction.It should be clear that this proof also yields that in Mitchell’s model, both Yω1 = Zω1and Yω2 = Zω2 for periodic FC-groups; especially Y = Z for groups of size ≤ω2.5.8. Theorem.

The consistency of ZFC + I implies the consistency of ZFC + thefollowing statements.29

(i) Yω1 = Zω1 for periodic FC-groups – and any periodic FC-group G with |G/Z(G)| =ω1 has an abelian subgroup A with [G : NG(A)] = ω1;(ii) Yω2 = Zω2 for periodic finite-by-abelian groups – and any periodic finite-by-abeliangroup G with |G/Z(G)| = ω2 has an abelian subgroup A with [G : NG(A)] = ω2;(iii) There is an FC-group G with |G/Z(G)| = ω2 but [G : NG(A)] ≤ω1 for all abelianA ≤G.Proof. Let κ be inaccessible in V. By [1, Theorem 8.8], there is a partial order IP suchthat for G IP-generic over V,V[G] |= MA + 2ω = ω2+ ”there are no weak Kurepa trees”.

(This is proved by a countable support iteration of length κ of partial orders which alter-natively make ω1-trees special and are ccc.) Now apply 4.4/4.5 (for (i)), 4.6 (for (iii)), and5.4 (for (ii)).As both MA and no weak Kurepa trees follow from the proper forcing axiom PFA (by[1, § 8] – see also [2, 7.10]), (i) through (iii) in the Theorem hold if we assume ZFC +PFA.5.9.

Proposition. For any cardinal κ ≥ω2 it is consistent that there is an extraspe-cial p-group of size κ such that for all maximal abelian subgroups A ≤G, [G : A] ≤ω1.Proof.

By the arguments of 5.2 and 5.3 it suffices to generically add a Kurepa treewith κ branches as follows (Folklore). Let IKκ := {p; p is a function and dom(p) = α × A,where α < ω1 and A ∈[κ]ω, and ran(p) ⊆2}, ordered by p ≤q iffα(p) ≥α(q) (whereα(p) is the α of the definition of the p.o.

), A(p) ⊇A(q), p⌈(α(q) × A(q)) = q, and for allβ ∈A(p) −A(q) there is γ ∈A(q) such that p⌈(α(q) × {β}) = p⌈(α(q) × {γ}). IKκ is (ifwe assume CH in the ground model V) ω2 −cc and ω1-closed, and so preserves cardinals.Clearly this works.30

§ 6. Generalizations6.1.

It was mentioned in the Introduction that (II) holds for arbitrary Zκ-groups G[4]. One might ask what goes wrong in this general case (where G is not required to beFC) if we drop the Zκ-condition.Proposition.

For any cardinal κ there is a group G with |G/Z(G)| = 2κ and [G :NG(A)] ≤κ for all abelian A ≤G.Proof. Let A and B be two elementary abelian p-groups of size κ.

Let h : [κ]2 →κbe a bijection. We define a factor system τ as follows:τ(aα, aβ) = 0if α ≥β,bh(α,β)otherwise,where the aα (bα, resp.) (α < κ) generate A (B, resp.

); extend τ bilinearly to A2. LetC := E(τ) be the extension.

(Note that C is the free object on κ generators in the varietyof two-step nilpotent groups of exponent p (p > 2); and that its maximal abelian subgroupsare of the form ⟨B, a⟩, where a ̸∈B.) Let D be a group of order p. Let G be the (abelian)subgroup of Aut(C ⊕D) consisting of all automorphisms φ which fix B ⊕D and satisfy∀c ∈C ⊕D ∃k ∈p (cφ = c + kd).Clearly, |G| = 2κ.

Let E be the semidirect extension of C⊕D and G (i.e. E = (C⊕D)×G).We leave it to the reader to verify that |E/Z(E)| = 2κ and [E : CE(A)] ≤κ for all abelianA ≤E.Note.

The proof is similar to (but easier than) the proof of Theorem A (see 4.2 and4.3). Unlike the latter it does not involve any set-theoretic hypotheses.

On the other hand,the group E constructed above is not κC but κ+C (a group G is κC iffevery g ∈G hasless than κ conjugates).6.2. We restricted our attention to κ = ω1 or ω2.

This is reasonable because theproblem seems to be most interesting for small cardinals. Also, the constructions in §5(5.3 and 5.9) show how to get consistency results concerning the existence of pathologicalgroups for larger cardinals (just use λ-Kurepa families instead of (ω1-)Kurepa families forthe appropriate λ).

Nevertheless we ignore whether the non-existence of such groups isconsistent for κ ≥ω3 (cf Theorem E).31

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